Related papers: Finite Element Representation Network (FERN) for O…

The ultimate goal of any numerical scheme for partial differential equations (PDEs) is to compute an approximation of user-prescribed accuracy at quasi-minimal computational time. To this end, algorithmically, the standard adaptive finite…

We analyze optimal complexity of adaptive finite element methods (AFEMs) for general second-order linear elliptic partial differential equations (PDEs) in the Lax-Milgram setting. To this end, we formulate an adaptive algorithm which steers…

Neural networks have shown significant potential in solving partial differential equations (PDEs). While deep networks are capable of approximating complex functions, direct one-shot training often faces limitations in both accuracy and…

Partial differential equations (PDEs) govern diverse physical phenomena, yet high-fidelity numerical solutions are computationally expensive and Machine Learning approaches lack generalization. While Scientific Foundation Models (SFMs) aim…

We present a new framework for computing fine-scale solutions of multiscale Partial Differential Equations (PDEs) using operator learning tools. Obtaining fine-scale solutions of multiscale PDEs can be challenging, but there are many…

Physics-informed neural networks (PINNs) [4, 10] are an approach for solving boundary value problems based on differential equations (PDEs). The key idea of PINNs is to use a neural network to approximate the solution to the PDE and to…

The finite element methods (FEM) are important techniques in engineering for solving partial differential equations, but they depend heavily on element shape quality for stability and good performance. In this paper, we introduce the…

We present a novel formulation for modeling phase-field fracture propagation based on the Integrated Finite Element Neural Network (IFENN) framework. IFENN is a hybrid solver scheme that utilizes neural networks as PDE solvers within FEM,…

Physics-Informed Neural Networks (PINNs) have emerged as a powerful framework for solving partial differential equations (PDEs) by embedding physical laws into neural network training. However, traditional PINN models are typically designed…

Finite element discretizations of problems in computational physics often rely on adaptive mesh refinement (AMR) to preferentially resolve regions containing important features during simulation. However, these spatial refinement strategies…

This article presents a three-step framework for learning and solving partial differential equations (PDEs) using kernel methods. Given a training set consisting of pairs of noisy PDE solutions and source/boundary terms on a mesh, kernel…

Physics-informed deep learning has emerged as a promising framework for solving partial differential equations (PDEs). Nevertheless, training these models on complex problems remains challenging, often leading to limited accuracy and…

This paper proposes an Adaptive Basis-inspired Deep Neural Network (ABI-DNN) for solving partial differential equations with localized phenomena such as sharp gradients and singularities. Like the adaptive finite element method, ABI-DNN…

We present a machine-learning strategy for finite element analysis of solid mechanics wherein we replace complex portions of a computational domain with a data-driven surrogate. In the proposed strategy, we decompose a computational domain…

Physics-Informed Neural Networks (PINNs) seek to solve partial differential equations (PDEs) with deep learning. Mainstream approaches that deploy fully-connected multi-layer deep learning architectures require prolonged training to achieve…

Spectral neural operators achieve strong performance for PDE learning, but rely on fixed global bases that limit their ability to represent spatially heterogeneous and multiscale dynamics. We propose Adaptive Basis Learning (ABLE), a…

Transfer learning for partial differential equations (PDEs) is to develop a pre-trained neural network that can be used to solve a wide class of PDEs. Existing transfer learning approaches require much information of the target PDEs such as…

To obtain fast solutions for governing physical equations in solid mechanics, we introduce a method that integrates the core ideas of the finite element method with physics-informed neural networks and concept of neural operators. This…

We propose an approach to solving partial differential equations (PDEs) using a set of neural networks which we call Neural Basis Functions (NBF). This NBF framework is a novel variation of the POD DeepONet operator learning approach where…

The approximation of solutions of partial differential equations (PDEs) with numerical algorithms is a central topic in applied mathematics. For many decades, various types of methods for this purpose have been developed and extensively…